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2020 Trees, dendrites and the Cannon–Thurston map
Elizabeth Field
Algebr. Geom. Topol. 20(6): 3083-3126 (2020). DOI: 10.2140/agt.2020.20.3083

Abstract

When 1HGQ1 is a short exact sequence of three word-hyperbolic groups, Mahan Mj (formerly Mitra) has shown that the inclusion map from H to G extends continuously to a map between the Gromov boundaries of H and G. This boundary map is known as the Cannon–Thurston map. In this context, Mj associates to every point z in the Gromov boundary of Q an “ending lamination” on H which consists of pairs of distinct points in the boundary of H. We prove that for each such z, the quotient of the Gromov boundary of H by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich and Lustig and Dowdall, Kapovich and Taylor, who prove that in the case where H is a free group and Q is a convex cocompact purely atoroidal subgroup of Out(FN), one can identify the resultant quotient space with a certain –tree in the boundary of Culler and Vogtmann’s Outer space.

Citation

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Elizabeth Field. "Trees, dendrites and the Cannon–Thurston map." Algebr. Geom. Topol. 20 (6) 3083 - 3126, 2020. https://doi.org/10.2140/agt.2020.20.3083

Information

Received: 25 July 2019; Revised: 22 January 2020; Accepted: 15 February 2020; Published: 2020
First available in Project Euclid: 16 December 2020

MathSciNet: MR4185936
Digital Object Identifier: 10.2140/agt.2020.20.3083

Subjects:
Primary: 20F65
Secondary: 20E07, 20F67, 57M07

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.20 • No. 6 • 2020
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